On the cardinality of irredundant and minimal bases of finite permutation groups

Abstract

AbstractGiven a finite permutation group G with domain $$\Omega $$ Ω , we associate two subsets of natural numbers to G, namely $${\mathcal {I}}(G,\Omega )$$ I ( G , Ω ) and $${\mathcal {M}}(G,\Omega )$$ M ( G , Ω ) , which are the sets of cardinalities of all the irredundant and minimal bases of G, respectively. We prove that $${\mathcal {I}}(G,\Omega )$$ I ( G , Ω ) is an interval of natural numbers, whereas $${\mathcal {M}}(G,\Omega )$$ M ( G , Ω ) may not necessarily form an interval. Moreover, for a given subset of natural numbers $$X \subseteq {\mathbb {N}}$$ X ⊆ N , we provide some conditions on X that ensure the existence of both intransitive and transitive groups G such that $${\mathcal {I}}(G,\Omega ) = X$$ I ( G , Ω ) = X and $${\mathcal {M}}(G,\Omega ) = X$$ M ( G , Ω ) = X .